On each corner of a square is a quarter. Your task is to have all four quarter heads-up or tails-up at the end of a turn.

You are blindfolded at the start, and you do not know which are heads-up and which are tails-up. Each turn, you may flip however many of them you want and then ask if you are done (and no, you cannot tell, by touch, whether it is heads- or tails-up). The square is then rotated a random, undisclosed number of quarter spins (multiple of 90 degrees), and you may take another turn.

Minimize the maximum number of turns required to be assured you will complete the task.

That first step would be unnecessary, if indeed you needed all heads-up
or tails-up at any point in the turn. But, as it states in the
second sentence, you must have four of a kind at the end of a
turn. If you start out in a HHHH situation, in your scenario, you
would never solve it, as after the first turn it would become
HTHT. Yours implies a buffer period, an 'Am I done' phase before
start of game, which, in the rules, happens only at the end of a
turn. Your scenario implies one turn of flipping nothing before
hand, as did the other solution.